It is proved that if
C
C
is a convex body in
R
n
{\mathbb {R}^n}
then
C
C
has an affine image
C
~
\tilde C
(of nonzero volume) so that if
P
P
is any
1
1
-codimensional orthogonal projection,
\[
|
P
C
~
|
≥
|
C
~
|
(
n
−
1
)
/
n
.
|P\tilde C| \geq \,|\tilde C{|^{(n - 1)\,/\,n}}.
\]
It is also shown that there is a pathological body,
K
K
, all of whose orthogonal projections have volume about
n
\sqrt n
times as large as
|
K
|
(
n
−
1
)
/
n
|K{|^{(n - 1)\,/\,n}}
.